Let $G$ be a locally compact abelian group and $\mathbb{T}$ the 1-torus, i.e. the unit circle in $\mathbb{C}$.

Definition - A continuous homomorphism $G\longrightarrow\mathbb{T}$ is called a character of $G$. The set of all characters is called the Pontryagin dual of $G$ and is denoted by $\hat{G}$.

Under pointwise multiplication $\hat{G}$ is also an abelian group. Since $\hat{G}$ is a group of functions we can make it a topological group under the compact-open topology (topology of convergence on compact sets).

• $\hat{\mathbb{Z}}\cong\mathbb{T}$, via $n\mapsto z^{n}$ with $z\in\mathbb{T}$.

• $\hat{\mathbb{T}}\cong\mathbb{Z}$, via $z\mapsto z^{n}$ with $n\in\mathbb{Z}$.

• $\hat{\mathbb{R}}\cong\mathbb{R}$, via $t\mapsto e^{{ist}}$ with $s\in\mathbb{R}$.

The following are some important properties of the dual group:

Theorem - Let $G$ be a locally compact abelian group. We have that

• $\hat{G}$ is also locally compact.

• $\hat{G}$ is second countable if and only if $G$ is second countable.

• $\hat{G}$ is compact if and only if $G$ is discrete.

• $\hat{G}$ is discrete if and only if $G$ is compact.

• $\widehat{(\oplus_{{i\in J}}G_{i})}\cong\oplus_{{i\in J}}\hat{G_{i}}$ for any finite set $J$. This isomorphism is natural.

Let $f:G\longrightarrow H$ be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map $\hat{f}:\hat{H}\longrightarrow\hat{G}$ defined by

This canonical construction preserves identity mappings and compositions, i.e. the dualization process $\hat{\;}$ is a functor:

Theorem - The dualization $\hat{\;}:\mathord{\mathbf{LcA}}\longrightarrow\mathord{\mathbf{LcA}}$ is a contravariant functor from the category of locally compact abelian groups to itself.

Although in general there is not a canonical identification of $G$ with its dual $\hat{G}$, there is a natural isomorphism between $G$ and its dual’s dual $\hat{\hat{G}}$:

Theorem - The map $G\longrightarrow\hat{\hat{G}}$ defined by $s\mapsto\hat{\hat{s}}$, where $\hat{\hat{s}}(\phi):=\phi(s)$, is a natural isomorphism between $G$ and $\hat{\hat{G}}$.

The study of dual groups allows one to visualize Fourier series, Fourier transforms and discrete Fourier transforms from a more abstract and unified view-point, providing the basis for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the foundations of the theory of abstract abelian harmonic analysis.